There is a difference between NATO and the UN. Membership in NATO is not open to Israel's enemies for one. Asking for NATO intervention means that if Syria or Iran want to enter the conflict they would have to go through NATO first. I think that this is a shrewd calculation by Israel and the United States.
There is only one nation on Earth that is not an enemy of Israel, and that is the US. And the US will not, and should not volunteer for this mission. It would simply be a target, as it was before.